Consider two neighboring points in \(\mathbb {P}\mathcal {T}\) .
These correspond to two twistor surfaces \(\Sigma , \Sigma ^{\prime }\)  in \(\mathbb {C}M\) , which we take to be associated
to the same congruence defined by the spinor field \(\alpha _{A}\) .
We now use the concept of a `connecting vector' as defined by Penrose and Rindler in [1]},
but slightly generalized to a congruence of surfaces instead of curves.
This is used by Ward and Wells in [2]}:
a vector field \(v\)  defined over \(\Sigma \)  is a `connecting' vector to the nearby twistor surface
\(\Sigma ^{\prime }\)  if the Lie derivative of \(v\)  along vectors tangent to \(\Sigma \)  is again tangent to \(\Sigma \) ,
i.e. \(£_{X}v\in \Gamma (T\Sigma )\)  for all \(X\in \Gamma (T\Sigma )\) .
Given the connecting vector \(v\) , the change in the spinor field \(\alpha _{A}\)  between the nearby
twistor surfaces \(\Sigma \)  and \(\Sigma ^{\prime }\)  is represented by the spinor field
\(\delta _{A} = v^{b}\nabla _{b}\alpha _{A}.\) 
